The Hot Big Bang Model and supporting evidence

I’m having a slightly hard time with a topic we are studying at school ‘the evolution of the universe’.

My first question relates to the Big Bang Model itself; does the Big Bang Model itself describe the evolution of the universe after the Planck time up to this point now? Secondly after 3×10^5 years my book states that ‘the universe becomes transparent so that em radiation in the form of photons moves freely through the universe’? I don’t really understand what it means for the universe to become ‘transparent’?

My second question relates to the evidence supporting the Big Bang theory.

A question in my book asks ‘state the features of cosmic microwave background radiation and explain how it provides evidence for the Big Bang theory’ , I’m not really quite sure about the evidence part, since I don’t understand the Big Bang all that well…but here is my attempt at an answer:

The features of cosmic microwave background radiation are that it corresponds to a temperature of 2.7K ( by corresponds does it mean it is at that temperature?! :S ) , the radiation can be detected in all directions in space and is almost perfectly uniform. According to the Big Bang Model , initially the universe was very hot and 3×10^5 years after the Big Bang electromagnetic radiation was able to move freely throughout the universe. In addition after the point of the Big Bang the universe began to expand and has been expanding every since, therefore these wavelengths must have been stretched out so that they are of a longer wavelength and therefore less energetic, thus corresponding to a lower temperature-2.7K. These waves were stretched all the way into the microwave region of the electromagnetic spectrum, hence the cosmic microwave background radiation that we detect today is the cooled remnants of the electromagnetic radiation that first started to travel around the universe 3×10^5 years after the Big Bang.